Local Basis

Relevant parts to questions...

• A local basis ${\mathbf{e}}_j={\mathbf{e}}_j(x^1,x^2,x^3)$ varies with position - so $d{\mathbf{e}}_j\ne 0$.
• The key differential identity: $d\mathbf{A}={\mathbf{e}}^{j}d{A}_j+A_{j}d{\mathbf{e}}^j={\mathbf{e}}_{j}d{A}^j+A^{j}d{\mathbf{e}}_j$.
• The extra basis-derivative term is what Christoffel Symbols package up.
• Fixed bases (Cartesian, rigid generalised) collapse $d{\mathbf{e}}_j=0$; local bases (spherical, cylindrical, curved surfaces) do not.

A Local Basis is a basis $\{{\mathbf{e}}_1(\mathbf{x}),{\mathbf{e}}_2(\mathbf{x}),{\mathbf{e}}_3(\mathbf{x})\}$ that depends on position. In cylindrical, spherical, or any curvilinear coordinate system, the basis vectors point in different directions at different points - so they carry position as an argument.

This is the key contrast with all earlier material:

• Fixed basis (e.g. Cartesian $\mathbf{i},\mathbf{j},\mathbf{k}$)::$d{\mathbf{e}}_j=0$. Differentials of vectors collapse to differentials of components.
• Local basis (e.g. cylindrical $\hat{\mathbf{\rho }},\hat{\mathbf{\varphi }},\hat{\mathbf{z}}$)::$d{\mathbf{e}}_j\ne 0$. The basis itself contributes to any derivative.

The Differential of a Vector Field

Expand $\mathbf{A}=A_j{\mathbf{e}}^j=A^j{\mathbf{e}}_j$. By the product rule:

$$\boxed{d\mathbf{A}={\mathbf{e}}^{j}d{A}_j+A_{j}d{\mathbf{e}}^j={\mathbf{e}}_{j}d{A}^j+A^{j}d{\mathbf{e}}_j}$$

Two terms instead of one. The first differentiates the components; the second accounts for the changing basis. Forgetting the second is the single most common mistake in Covariant Differentiation.

Partial Derivatives

Since $d\mathbf{A}=\frac{\partial \mathbf{A}}{\partial x^k}d{x}^k$, the partial derivative inherits the same two-term structure:

$$\frac{\partial \mathbf{A}}{\partial x^k}=\frac{\partial A_j}{\partial x^k}{\mathbf{e}}^j+A_j\frac{\partial {\mathbf{e}}^j}{\partial x^k}=\frac{\partial A^j}{\partial x^k}{\mathbf{e}}_j+A^j\frac{\partial {\mathbf{e}}_j}{\partial x^k}$$

The basis-derivative term $\frac{\partial {\mathbf{e}}_j}{\partial x^k}$ is the geometric fingerprint of the local basis - and is precisely what the Christoffel Symbols ${\Gamma }_{jk}^i$ package as coefficients in the basis expansion $\frac{\partial {\mathbf{e}}_j}{\partial x^k}={\Gamma }_{jk}^i{\mathbf{e}}_i$.

Properties

• Derived from a parametrisation::given $\mathbf{r}(x^1,x^2,x^3)$, the local basis is ${\mathbf{e}}_i=\frac{\partial \mathbf{r}}{\partial x^i}$.
• Metric Tensor inherits position-dependence::$g_{ij}(\mathbf{x})={\mathbf{e}}_i(\mathbf{x})\cdot {\mathbf{e}}_j(\mathbf{x})$ is now a field, not a constant.
• Fixed is a special case::if $\frac{\partial {\mathbf{e}}_j}{\partial x^k}=0$ everywhere, all Christoffel symbols vanish and covariant differentiation collapses to ordinary partial differentiation.

Applications

1. Physically-motivated coordinates (spherical, cylindrical, oblique) all use local bases.
2. Curved manifolds (the surface of a sphere, hyperbolic plane) only admit local bases - no rigid basis can cover them.
3. Setting up covariant differentiation correctly - the two-term differential is the starting point for every proof in Chapter 7.

Cylindrical basis varies with $\theta$

${\mathbf{e}}_1=\cos \theta {\mathbf{i}}_1+\sin \theta {\mathbf{i}}_2$ (radial), so $\frac{\partial {\mathbf{e}}_1}{\partial \theta }=-\sin \theta {\mathbf{i}}_1+\cos \theta {\mathbf{i}}_2\ne 0$. The radial basis genuinely rotates as you move around the $z$-axis - a local basis in action. $\boxed{\frac{\partial {\mathbf{e}}_1}{\partial \theta }\ne 0}$ ✓

Fixed basis: basis derivatives vanish. Local basis: basis derivatives are the story.